Journal of Engineering Mathematics最新文献

We first study the existence and uniqueness of a positive self-similar solution of the 2D boundary-layer equations of an incompressible viscous power-law fluid when the external flow is accelerating, and then we derive the bounds of the wall shear stress rate. For shear-thickening fluids, we show that the matching with the external flow occurs at a finite distance. Furthermore, we also investigate the asymptotic behaviour at infinity of positive solutions in the case of shear-thinning fluids.

In this paper, we present a mathematical study of particle diffusion inside and outside a spherical biological cell that has been exposed on one side to a propagating planar diffusive front. The media inside and outside the spherical cell are differentiated by their respective diffusion constants. A closed form, large-time, asymptotic solution is derived by the combined means of Laplace transform, separation of variables, and asymptotic series development. The solution process is assisted by means of an effective far-field boundary condition, which is instrumental in resolving the conflict of planar and spherical geometries. The focus of the paper is on a numerical comparison to determine the accuracy of the asymptotic solution relative to a fully numerical solution obtained using the finite element method. The asymptotic solution is shown to be highly effective in capturing the dynamic behaviour of the system, both internal and external to the cell, under a range of diffusive conditions.

Two two-dimensional free boundary problems describing the erosion of solid surfaces by the flow of inviscid fluid in the presence of trapped vortices are considered. The first problem tackles an initially flat, infinite fluid-solid interface with uniform flow at infinity and a vortex in equilibrium above the surface. The second involves flow around a finite body with a trailing Föppl-type vortex pair. The conformal invariance of the complex potential permits both problems to be formulated as a Polubarinova–Galin (PG) type equation in which the time-dependent eroding surface in the physical z-plane is mapped to the fixed boundary of the (zeta )-disk. The Hamiltonian governing the equilibrium position of the vortex (or vortex pair in the second problem) is also found from the same map. In each problem, the PG equation giving the conformal map is found numerically and the time-dependent evolution of the interface and vortex location is determined. Different models governing the erosion of the interface are investigated in which the normal velocity of the boundary depends on some given function of the fluid flow velocity at the boundary. Typically, in the infinite surface case, erosion leads to the formation of a symmetric valley beneath the vortex which, in turn, moves downward toward the interface. A finite body undergoes erosion which is asymmetric in the flow direction leading to a flattening of the lee surface of the body so displaying some similarity to the experiments and associated viscous theory of Ristroph et al, Moore et al (Proc Natl Acad Sci 109(48):19606–19609, 2012, Phys Fluids 25(11):116602, 2013).

Spreading of liquid drops on cold solid substrates is a complicated problem that involves heat transfer, fluid dynamics, and phase change physics combined with complex wetting behaviors at the contact line. Understanding the physics behind the non-isothermal spreading of droplet is of utmost importance due to its broad applications in diverse areas of industry such as in additive manufacturing processes. This work mainly focuses on determining the important physical parameters involved in the non-isothermal spreading of droplets with low contact angle ((theta <pi /2)) as well as controlling the post-solidification geometry of impinging droplets with moderate impact velocity where spreading is driven by impact velocities, but fingerings or instabilities do not occur at the contact line. Using analytical modeling, a possible explanation for contact-line arrest is produced that demonstrates that the final radius of droplets of moderate impacting velocity is independent of the initial conditions including the impact dynamics and temperature gradients. The predictive capacity of this model is confirmed with experimental results.

This study implemented a novel technique to address the common issue of stripe pattern formation in 2D systems known as the time-fractional Newell–Whitehead–Segel problem. The study presents the Mohand transforms and their properties in conformable sense. The proposed solution involved utilizing the homotopy perturbation approach (HPA) and conformable Mohand transform (CMT) to tackle four case studies of the time-fractional Newell–Whitehead–Segel problem. The graphical outcomes produced by the suggested approach resembled the exact solution. The effectiveness of the suggested techniques was demonstrated by presenting precise and analytical data through graphs. Additionally, the results of using the suggested technique for different values of (alpha ) were compared, showing that as the value moves from a fractional order to an integer order, the answer becomes more and more similar to the exact solution.

This paper investigates the dynamic behavior of a V-notch with non-trivial boundaries in a piezoelectric/piezomagnetic half-space. We start by considering a SH wave impinging on the piezoelectric/piezomagnetic half-space. Upon employing the superposition principle, an expression for the scattering wave is derived, which meets the required conditions at the boundary of the half-space. Subsequently, we provide the analytic expression for the standing wave, formulated to meet the stress-free assumptions and electric/magnetic insulation at the boundaries of the V-notch. This is done using an expansion in fractional Bessel functions and the Graf theorem. Finally, a method based on Green’ functions is employed to divide the half-space along the vertical interface, where in-plane electric and magnetic fields and out-of-plane forces are exerted. This leads to the formulation of integral Fredholm equations, which are solved using an expansion into orthogonal functions and an effective truncation technique. Our results describe the scattering effect on the concentration factors of the dynamic stress, and of electric and magnetic fields in relevant conditions. The analytic solutions are validated using finite element method, and results confirm the accuracy of our findings.

An analytical investigation is conducted to analyze the impact of magnetic field and hydrodynamic slippage on two-dimensional electro-osmotic Brinkman flow in a microchannel with cosine surface zeta potential. The Brinkman equation is utilized to govern the fluid flow within a fully saturated, homogeneous, and isotropic porous medium. We consider a very small magnetic Reynolds number to eliminate the induced magnetic field equation. The Navier slip boundary condition is applied to assess the impact of hydrodynamic slippage. We utilize the Debye–Huckel length approximation to linearize the Poisson–Boltzmann equation, which governs the potential of the electrical double layer. The stream function is obtained analytically, and contour plots, velocity fields, shear stresses, and pressure gradients are assessed to gain a proper understanding of flow physics. We utilize the stream function to plot the streamline plots for distinct assumed flow parameters. We observed that for a fixed Darcy number, the intensity of flow vortices decreases with increasing Hartman number while increasing with increasing slip length. Further, altering the wave number in the assumed cosine-waved zeta potential causes asymmetrical recirculations in the flow, which helps in increasing the scalar mixing process in microdevices. Further, the proposed investigation has various crucial applications, such as microfluidic cooling systems, drug delivery systems, and so on.

When a cancerous cell colony grows within a healthy environment, the entire structure can be modelled as a continuous two-phase fluid with five bounded compartments, governed by the laws of mass conservation, Fick’s diffusion law, and fluid mechanics principles. The interfaces of the five bounded compartments of the colony are defined by critical values of nutrient concentration. In studying the evolution of the exterior tumour boundary, nutrient concentration is the primary parameter. Although most existing research focuses on spherical tumours, significant implications for nutrient distribution emerge when spherical symmetry is abandoned, such as the occurrence of critical values at specific points rather than across the entire surface. In this work, we consider an oblate spheroidal tumour and investigate the effects of non-homogeneity in both nutrient supply and consumption rates. Our findings indicate that critical values are encountered within the interior of a thin layer, rather than at a single interface, although the interface is still included. We study the variation of nutrient concentration on the tumour’s interfaces through plots, highlighting the critical locations. The prolate spheroidal case can be derived via a simple transformation, and comparisons with similar spherical models are also discussed.

Electrokinetic flows driven by electro-osmotic forces are especially relevant in micro and nano-devices, presenting specific applications in medicine, biochemistry, and miniaturized industrial processes. In this work, we integrate analytical solutions with numerical methodologies to explore the fluid dynamics of viscoelastic electro-osmotic/pressure-driven fluid flows (described by the generalized Phan–Thien–Tanner (gPTT) constitutive equation) in a microchannel under asymmetric zeta potential conditions. The constitutive equation incorporates the Mittag–Leffler function with two parameters ((alpha ) and (beta )), which regulate the rate of destruction of junctions in a network model. We analyze the impact of the various model parameters on the velocity profile and observe that our newly proposed model provides a more comprehensive depiction of flow behavior compared to traditional models, rendering it suitable for modeling complex viscoelastic flows.

A Hybrid Potential Flow (HPF) model for flow around a circular cylinder in the subcritical Reynolds number range ((300 le Re le 3times 10^5)) is developed using a combination of elementary flow solutions and empirical data. By joining this developed near-body solution with von Karman’s model for the vortex wake, a complete solution for flow around a circular cylinder is calculated. Results for oscillatory forces, including the transverse lift force, due to vortex shedding as well as shedding frequencies are then calculated and presented. With the complete solution for flow around a cylinder calculated, the HPF model can be used as a step to calculate the flow around other bluff bodies using conformal mapping, an approach that has been developed and presented by the authors in a related paper.